Problem: What is the intersection of the lines given by $y=-4x$ and $y-2=12x$? Express your answer as an ordered pair, with both coordinates expressed as common fractions.
To find the intersection, we must find the point satisfying both equations. Hence we must solve the system \begin{align*}
y&=-4x, \\
y-2&=12x.
\end{align*}Substituting the expression for $y$ in first equation into the second equation, we obtain $-4x-2=12x$. Solving for $x$ we find that $x=-\frac{1}{8}$. Plugging this into the first expression for $y$ above, we find that $y=-4\cdot -\frac{1}{8}=\frac{1}{2}$. So the intersection is $\boxed{\left(-\frac{1}{8}, \frac{1}{2}\right)}$.